HOW CAN (AND WHY SHOULD) PD DRIVE
MATHEMATICAL JUSTIFICATION AND GENERALIZATION AS
CLASSROOM NORMS? AND HOW DO WE KNOW WHEN IT
DOES?Julie Fredericks
Teachers Development Group
Session Overview Definition and Purpose
What is a mathematical justification and what purposes does mathematical justification serve in the K-12 classroom?
Challenges & Strategies What are the challenges that the teachers we
work with face regarding mathematical justification?
Specific examples of tools and strategies we use in our work
Our Wonderings and Future Plans
Mathematical DiscourseAn evidence-based premise of the studio work is the notion that orchestrating productive mathematical discourse increases students’ opportunities to learn and, in turn, raises achievement and participation levels in mathematics.
Embracing this premise requires developing teachers’ knowledge, skills, tools, and disposition for building classroom communities of mathematical discourse. Leahy, Lyon, Thompson, and Wiliam, 2005; Yackel & Cobb, 1996; Hufferd-Ackles & Sherin, 2004; Stein, Engle, Hughes & Smith, 2008; Weaver & Dick, 2006
Driven by how students learn mathematics
Student mathematics achievement will improve if teachers consistently use research-based instructional practices to develop both computational fluency and a deep understanding of mathematics concepts by engaging all students consistently and effectively in the following mathematical practices:
Providing Explanations Making Justifications Formulating Conjectures & Generalizations. Using Multiple Representations Engaging in Metacognition Making Connections Bransford et al, 1999; Cohen, 1994; Donovan & Bransford,
2005; Franke et al, 2007; Kilpatrick, 2001; Lotan, 2003, 2006; Stein et al, 2000; Common Core State Standards Initiative, 2010
Let’s Do Some MathStructuring the discussion. Start with 5 minutes private think time. Find a partner at your table and switch
papers. Review your partners work WITHOUT talking.
First partner shares their interpretation of other partners work. Other partner clarifies.
Second partner shares their interpretation of first partners work. First partner clarifies.
Both discuss compare and contrast their solutions.
Let’s Do Some MathProve that the sum of two odd numbers is
even.
After you have solved the problem, think about what is mathematical justification? At which points in your work have you provided mathematical justification.
Let’s Do Some MathStructuring the discussion. Start with 5 minutes private think time. Find a partner at your table and switch
papers. Review your partners work WITHOUT talking.
First partner shares their interpretation of other partners work. Other partner clarifies.
Second partner shares their interpretation of first partners work. First partner clarifies.
Both discuss compare and contrast their solutions.
Purpose What are some of the possible purposes
that justification could serve in the K-12 mathematics classroom?
Common Core Standards“One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.”
-Common Core State Standards for Mathematics
Common Core StandardsStandards for Mathematical Practice1. Make sense of problems and persevere in
solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the
reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated
reasoning.
Challenges & Strategies Teachers recognizing mathematical
justification. Students recognizing mathematical
justification. Choosing when to press for mathematical
justification.
Challenges & StrategiesTeachers recognizing mathematical
justification Teachers often struggle with the
differences between mathematical justification, explanation, and verification.
Challenges & StrategiesProviding opportunities for teachers to discuss
their wonderings and questions about mathematical justification.
“[I am] still trying to understand what justify means in the context of 8th grade algebra vs. a general problem solving task vs. high school geometry. Not always sure when students are verifying vs. justifying.”
“First grade has been working on word problems. We want our students to be able to solve a word problem and show how they go their answer instead of just the answer itself, is this justification?”
Challenges & StrategiesStudent Mathematical Discourse
Observation Tool Watch a video clip of a 3rd grader. As we watch, we are going to code the
student discourse. Particularly with an eye for justification, conjectures and generalizations.
Allison Video from Thinking Mathematically: Integrating Arithmetic and Algebra
in Elementary School (Carpenter, Franke, and Levi, 2003)
Use Student Discourse Observation Protocol
5 minutes PTT – Use the provided transcript of the video to code the student mathematical discourse from this episode.
At your table, share one place that you identified a justification, conjecture or generalization.
Challenges & Strategies Students recognizing mathematical
justification. When first pressing for justification in the
classroom, students have trouble understanding what you want.
A Looks Like/ Sounds Like Chart for Mathematical Justification from a classroom
Challenges & StrategiesStructured Student Math Talk
Using the structure to press students to share and compare their thinking .
Today we used the Interpret and Compare routine which is outlined on the second page of the tool.
Challenges & Strategies Knowing when to press for justification
Deciding which questions are appropriate for grade level “I am finding that my students can explain how
they solved a math problem, but I am stumbling when it comes to explaining why (justification). Right now we have a standing question in our class: Can you generalize a rule about how diameter and circumference are related and justify why your rule would work with any circle.”
Pressing for justification on every question regardless of mathematical goal of lesson.
Challenges & Strategies Spend explicit time during lesson
planning for mathematical justification. For which questions would pressing for
justification be most important? Why? What does are the acceptable justifications for
this task?
Planning Questions
Challenges & Strategies “It seems as though this is starting to become one of the
norms in our math conversations. I can think of at least 4-5 different occasions in collaborative meetings, with different teachers, that someone said "what would you want kids to do/show for their justification?" This seems to be becoming more of a norm.
It is a simple, yet complex, question and really forces the teacher to think ahead of time about what students will need to know and what will be "acceptable justification." Now, if we can all calibrate our own definitions of "acceptable justification" we will really move forward as a school. This is where I think it is necessary for us to continue to work together, refine our ideas in order to come to these shared understandings. This cannot be done in isolation. “
Our Wonderings and Future Plans
Continuum of justification How can we evaluate teachers growth? How can teachers evaluate their students
growth?
Thank You!!